Mathematical Problems in Engineering Volume, Article ID 769, pages doi:.55//769 Research Article Numerical Integration and Synchronization for the -Dimensional Metriplectic Volterra System Gheorghe Ivan, Mihai Ivan, and Camelia Pop Seminarul de Geometrie şi Topologie, West University of Timişoara, 4, B-dul V. Pârvan, Timişoara, Romania Math Department, The Politehnica University of Timişoara, Piaţa Victoriei nr., 6 Timişoara, Romania Correspondence should be addressed to Gheorghe Ivan, ivan@math.uvt.ro Received January ; Accepted 9 June Academic Editor: Marcelo Messias Copyright q Gheorghe Ivan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The main purpose of this paper is to study the metriplectic system associated to -dimensional Volterra model. For this system we investigate the stability problem and numerical integration via Kahan s integrator. Finally, the synchronization problem for two coupled metriplectic Volterra systems is discussed.. Introduction To give a unification of the conservative and dissipative dynamics, Kaufman has introduced the concept of metriplectic system. Let x,x,...,x n be a local coordinate system on R n. We consider ẋ t P x t H x t. be a Hamilton-Poisson system on R n with Hamiltonian H C R n, where x t x t,...,x n t T and H x H/ x,..., H/ x n T. We add to the Hamilton-Poisson system. a dissipation term of the form G x t C x t, where G x is a symmetric matrix which satisfies certain compatibility conditions, and C x t a C x t, where a R and C C R n are a Casimir function i.e., P x C x. One obtains a family of metriplectic systems of the form ẋ t P x t H x t G x t C x t..
Mathematical Problems in Engineering This family of metriplectic systems have the same Hamiltonian H and the same Casimir function C. For each a R, the metriplectic systems. can be viewed as a perturbation of Hamilton-Poisson system.. The differential systems of the form. and their applications have been studied in connection with several dynamical systems derived from mathematical physics; see for instance, 4. Another way for giving rise to a dynamical system of the form of. is based on the definition of a metriplectic structure on R n. These systems can be expressed in terms of Leibniz bracket, see 5 8. The paper is structured as follows. In Section, the metriplectic system associated to -dimensional Volterra model.8 is constructed. For this dynamical system, the stability of equilibrium states is investigated. In Section, we discuss the numerical integration for the system.8. Synchronization problem for dynamical systems has received a great deal of interest duetotheirapplicationindifferent fields of science; see 9. For this reason, Section 4 is dedicated to synchronization problem for two coupled metriplectic Volterra systems of the form of.8.. The Metriplectic System Associated to -Dimensional Volterra Model Let R n,p,h be a Hamilton-Poisson system given by.. For this system we determine the symmetric matrix G G ij, where G ii x n k,k/ i ( H x k ), G ij x H x i H x j, for i / j.. If C C R n is a Casimir function for the configuration R n,p,h, then we take C a C, where a R is a parameter. For P, H, C, and the matrix G determined by relations., we write the differential system. in the following tensorial form: ij H ẋ i P G ij C, i,j,n.. x j x j System. is a metriplectic system in R n see, 7, that is the following conditions are satisfied: i G x H x ; ii ( C x ) T G x C x.. System. is called the metriplectic system associated to Hamilton-Poisson system. andisdenotedby R n,p,h,g, C. Let us construct a metriplectic system of the form of., starting a Hamilton-Poisson realization of the -dimensional Volterra model.
Mathematical Problems in Engineering The phase space of the -dimensional Volterra model consists of variables x i, i ; see, 4. This is described by the equations ẋ x x, ẋ x x x x, ẋ x x..4 It is well known that system.4 has a Hamilton-Poisson realization R,P V,H V with the Casimir C V C R, R see 4, where x x P V x x x x, x x H V x,x,x x x x, C V x,x,x x x..5.6 We apply now relations. to the function H H V C R, R given by.6. Then the symmetric matrix G : G V G ij V is G V..7 We take H H V and C C V given by.6, the skew-symmetric matrix P P V given by.5 and the symmetric matrix G V given by.7. For the function C V ac V with a R, system. becomes ẋ x x a x x, ẋ x x x x a x x,.8 ẋ x x a x x. Proposition.. The dynamical system R,P V,H V,G V, C V given by.8 is a metriplectic system on R. Proof. We have C V / x x,and C V / x, C V / x x. Then x x x P V C V x x x x,.9 x x x that is, C V is a Casimir of Hamilton-Poisson system R,P V,H V. We prove that conditions i and ii from. are verified.
4 Mathematical Problems in Engineering We have C V x ax x, H V / x H V / x H V / x and C V / x ax, C V / x, C V / x ax. Then G V H V, ax ( T ) C x ) G x C x ax,,ax a( x x x x. ax. Hence.8 is a metriplectic system. System.8 is called the -dimensional metriplectic Volterra system. For a, it is reduced to Volterra model.4. System.8 can be written in the form ẋ i f i x, i,, where f x x x a x x, f x x x x x a x x, f x x x a x x.. Proposition.. i The function H V given by.6 is a constant of the motion for the metriplectic Volterra system, that is, it is conserved along the solutions of the dynamics.8. ii The function C V decreases along the solutions of system.8. Proof. i We have dh V /dt ẋ ẋ ẋ f x f x f x. ii The derivative of C V along the solutions of system.8 verifies the condition d C V /dt. Indeed, d C V /dt aẋ x ax ẋ a x f x x f x a x x x. x Remark.. If a /, then C V.8. ac V is not a constant of motion for the metriplectic system Proposition.4. i If a R, then the equilibrium states of the dynamics of.8 are e M,M, for all M R. ii) For a, the equilibrium states of the dynamics of.8 are e MN M,,N, and e M,M,, e M,,M for all M, N R. Proof. The equilibria are the solutions of system f i x, i,.
Mathematical Problems in Engineering 5 Proposition.5. The equilibrium states e M, M R are unstable. Proof. Let A be the matrix of the linear part of the system.8, thatis, x a x a A x a x x x a.. a x x a The characteristic polynomial of the matrix A e M is p λ λ λ aλ a M with the roots λ, λ, a ± 4a M. Then the assertion follows via Lyapunov s theorem 5. Remark.6. i The dynamics of.4 and.8 have not the same equilibria. ii For a, e MN, e M,eM have the following behaviors see 4 : e MN is unstable if M N<and spectrally stable if M N>; e M is unstable, and e M is unstable if M and spectrally stable if M<.. Numerical Integration of the Metriplectic Volterra System.8 For.8, Kahan s integrator see for details 6 can be written in the following form: x n x n h ( ) ( ) x n x n xn x n ah x n xn x n xn, x n x n h ( ) ( ) x n x n xn x n xn x n xn x n ah x n xn x n xn, ( ) ( ) x n x n h x n x n xn x n ah x n xn x n xn.. Remark.. Taking a in relations. we obtain the numerical integration for Volterra model.4 via Kahan s integrator. Proposition.. Kahan s integrator. preserves the constant of motion H V of the dynamics of.8. Proof. Indeed, adding all equations. we obtain x n x n x n x n xn xn.. Hence H V x n,x n,x n H V x n,xn,xn. For the initial conditions x, x, and x, the solutions of Volterra model.4 using Kahan s integrator. with a, are represented in the system of coordinates Ox x x in Figure. For the same initial conditions, the solutions of the metriplectic Volterra system.8 for a using Kahan s integrator. with a, are represented in the system of coordinates Ox x x in Figure.
6 Mathematical Problems in Engineering y 4 6 x.8 z.4 Figure : Kahan s integrator for Volterra model.4. 48 6 4 6 4 Figure : Kahan s integrator for the metriplectic Volterra system.8 with a. Remark.. Using Runge-Kutta 4 steps integrator, we obtain almost the same result; see Figure. 4. The Synchronization of Two Metriplectic Volterra Systems In this section we apply Pecora and Carroll method for constructing the drive-response configuration see. Let us build the configuration with the drive system given by the metriplectic Volterra system.8, and a response system this is obtained from.8 by replacing x i with y i and adding u i for i,. Suppose that these systems are coupled. More precisely, the second system is driven by the first one, but the behavior of the first system is not affected by the second one.
Mathematical Problems in Engineering 7 y.5.5 x z Figure : Runge-Kutta s integrator for Volterra model.4. 5 5 5 4 Figure 4: Runge-Kutta s integrator for the metriplectic Volterra system.8 with a. Therefore, the drive and response systems are given by ẋ x x a x x, ẋ x x x x a x x, 4. ẋ x x a x x,
8 Mathematical Problems in Engineering 5 8 6 4 4 5 6 4 5 6.5.5.5.865.86.855 4 5 6 a b c Figure 5: Synchronization of systems 4. and 4. for a. The solutions x t,y t and the evolution of error e t..6.5.4....5.5.5 5 8 6 4 4 5 6 a 4 5 6 b 4 5 6 c Figure 6: Synchronization of systems 4. and 4. for a. The solutions x t,y t and the evolution of error e t. respectively ẏ y y a ( y y ) u, ẏ y y y y a ( y y ) u, 4. ẏ y y a ( y y ) u, where u t,u t, and u t are three control functions. We define the synchronization error system as the subtraction of the metriplectic Volterra model 4. from the controlled metriplectic Volterra model 4. : e t y t x t, e t y t x t, 4. e t y t x t.
Mathematical Problems in Engineering 9 6. 4 5 6.4.6.8..4 4 5 6.5.5.4... 4 5 6 a b c Figure 7: Synchronization of systems 4. and 4. for a. The solutions x t,y t and the evolution of error e t. By subtracting 4. from 4. and using notation 4. we can get ė e e x a e x e ae u, ė e e e e a x e x x e a x e u, 4.4 ė e e ae x e a x e u. We define the active control inputs u t,u t, and u t as follows: u t e t e t e t u t K e t e t e t e t e t, 4.5 u t e t e t e t where k k k K k k k k k k 4.6 and k ij, i, j, are real functions which depend on x t,x t, and x t. Then the differential system of errors 4.7 is given by ė e e x a k e x k e k a e, ė e e e e a x k e x x k e a x k e, 4.7 ė e e k a e k x e a x k e. If we choose a x x K x a x x x, 4.8 a x x a x
Mathematical Problems in Engineering then the active controls defined by 4.5 become ( ) u e e a x x e, ( ) u e e e e x a e x x x e, ( ) u e e ae x e a x x e. 4.9 Using 4.8, the system of errors 4.7 becomes ė x e x e ae, ė x e a x e, 4. ė x e. Proposition 4.. The equilibrium state,, of the differential system 4. is asymptotically stable. Proof. An easy computation shows that the all conditions of Lyapunov-Malkin theorem 7 are satisfied, and so we have that the equilibrium state,, is asymptotically stable. Numerical simulations are carried out using the software MATHEMA-TICA 6. We consider the case a. The fourth-order Runge-Kutta integrator is used to solve systems 4., 4.,and 4. with the control functions u t,u t,u t given by 4.9. The initial values of the drive system 4. and response system 4. are x, x, x andy, y, y. These choices result in initial errors of e., e., and e.. The dynamics of the metriplectic Volterra system 4. to be synchronized with the dynamic of 4. accompanied with the control functions given by 4.9 and the dynamics of synchronization errors given by 4. are shown in Figures 5, 6, and7. According to numerical simulations, by a good choice of parameters the synchronization error states e t,e t,e t converge to zero, and hence the synchronization between two coupled metriplectic Volterra systems is achieved. Remark 4.. Taking a in 4., 4., 4.8, and 4.9, we obtain the synchronization between two coupled Volterra models of the form of.4. 5. Conclusion It is well known that many nonlinear differential systems like the Euler equations of fluid dynamics, the soliton equations can be written in the Hamiltonian form. An interesting example of nonlinear lattice equations is Volterra lattice see 8 which is a model for vibrations of the particles on lattices. Also the behavior of viscoelastic materials is an example where the dynamics is governed by Volterra equations. The metriplectic systems will be successfully used in mathematical physics, fluid mechanics, and information security; see for instance 4, 5,,.
Mathematical Problems in Engineering In this paper we have build a metriplectic system on R associated to Volterra model. For the metriplectic Volterra system.8, we have presented some relevant geometrical and dynamics properties and the numerical integration. Finally, using the Pecora and Carroll method, the synchronization problem for two coupled metriplectic Volterra systems of the form of.8 is discussed. This technique is realized since a suitable control has been chosen to achieve synchronization. References A. N. Kaufman, Dissipative Hamiltonian systems: a unifying principle, Physics Letters A, vol., no. 8, pp. 49 4, 984. P. Birtea, M. Boleanţu, M. Puta, and R. M. Tudoran, Asymptotic stability for a class of metriplectic systems, Mathematical Physics, vol. 48, no. 8, Article ID 87, 7 pages, 7. M. Grmela, Bracket formulation of dissipative fluid mechanics, Physics Letters A, vol., no. 8, pp. 55 58, 984. 4 P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physica D, vol. 8, no., pp. 4 49, 986. 5 D. Fish, Geometrical structures of the metriplectic dynamical systems, Ph.D. thesis, Portland, Ore, USA, 5. 6 P. Guha, Metriplectic structure, Leibniz dynamics and dissipative systems, Mathematical Analysis and Applications, vol. 6, no., pp. 6, 7. 7 G. Ivan and D. Opriş, Dynamical systems on Leibniz algebroids, Differential Geometry Dynamical Systems, vol. 8, pp. 7 7, 6. 8 J.-P. Ortega and V. Planas-Bielsa, Dynamics on Leibniz manifolds, Geometry and Physics, vol. 5, no., pp. 7, 4. 9 H. N. Agiza and M. T. Yassen, Synchronization of Rössler and Chen chaotic dynamical systems using active control, Physics Letters A, vol. 78, no. 4, pp. 9 97,. O. T. Chiş and D. Opriş, Synchronization and cryptography using chaotic dynamical systems, in Proceedings of the International Conference of Differential Geometry and Dynamical Systems, vol. 6 of BSG Proceedings [BGSP], pp. 47 56, Geometry Balkan Press, Bucharest, Romania, 9. M.-C. Ho, Y.-C. Hung, and I.-M. Jiang, On the synchronization of uncertain chaotic systems, Chaos, Solitons & Fractals, vol., pp. 54 546, 7. L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters, vol. 64, no. 8, pp. 8 84, 99. L. D. Fadeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, Germany, 986. 4 M. Puta and M. Khashan, Poisson geometry of the Volterra model, Analele Universităţii din Timişoara, Seria Matematică Informatică, vol. 9, pp. 77 84,. 5 M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, NY, USA, 974. 6 W. Kahan, Unconventional numerical methods for trajectory calculations, Unpublished lecture notes, 99. 7 D. V. Zenkov, A. M. Bloch, and J. E. Marsden, The energy-momentum method for the stability of non-holonomic systems, Dynamics and Stability of Systems, vol., no., pp. 65, 998. 8 Y. B. Suris, Integrable discretizations for lattice system: local equations of motion and their Hamiltonian properties, Reviews in Mathematical Physics, vol., no. 6, pp. 77 8, 999.
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